Every theme needs a the wp_head() and wp_footer() tags or plugins may not function as designed. These tags are frequently used by plugins to add javascript, stylesheets, and meta tags.
Try looking for, and if it is missing, add <?php wp_head(); ?> right before </head> in the header.php file in your theme directory.
Try looking for, and if it is missing, add <?php wp_footer(); ?> right before </body> in the footer.php file in your theme directory.

To easily confirm if the theme you are using is the cause:

Do this as a test:
In Admin, click on Appearance, Themes.
Temporarily activate your theme to a WP default one…
“WordPress Default” theme(default for WP 2.0+), or
“Twenty Ten” (default for WP 3.0+), or
“Twenty Eleven” (default for WP 3.2+) or
“Twenty Twelve” (default for WP 3.5+)
“Twenty Thirteen” (default for WP 3.6+)

It does not cause any harm to your WP to temporarily change the theme, test if the errors went away, and then and change the theme back.
Does it work properly with a default WP theme?
If it does, then the theme you are using is the cause. Use a different theme, or fix the missing wp_head() or wp_footer() tags.
You should also report the missing tags to the theme developer.

Solution – PHP version:
The captcha refresh button may not be working because your could be could be missing the contact-form.js JavaScript file. You are instructed to add this file by Step 5 of the readme.txt installation instructions included in the download package. The solution – this step is required for the CAPTCHA refresh to work properly:
Put this HTML code in your page somewhere in the head section anywhere after <head> but before </head>

Make sure you have replaced “www.yourwebsite.com” with your own web site, and if you have installed /contact-files/ in a sub directory you have added the path also. You should be able to put http://www.yourwebsite.com/contact-files/contact-form.js in a web browser and not get a “not found message”. And after installing, be sure to test that the CAPTCHA refresh button works, if it does not work you have performed this step incorrectly.